1. Superconducting properties of carbon nanotubes
Reinhold Egger
Institut für Theoretische Physik
Heinrich-Heine Universität Düsseldorf
A. De Martino, F. Siano

2. Overview Superconductivity in ropes of nanotubes
Attractive interactions via phonon exchange
Effective low energy theory for superconductivity
Quantum phase slips, finite resistance in the superconducting state
Josephson current through a short nanotube
Supercurrent through correlated quantum dot via Quantum Monte Carlo simulations
Kondo physics versus π-junction, universality

3. Classification of carbon nanotubes
Single-wall nanotubes (SWNTs):
One wrapped graphite sheet
Typical radius 1 nm, lengths up to several mm
Ropes of SWNTs:
Triangular lattice of individual SWNTs (typically up to a few 100)
Multi-wall nanotubes (MWNTs):
Russian doll structure, several inner shells
Outermost shell radius about 5 nm

4. Superconductivity in ropes of SWNTs: Experimental results
Kasumov et al., PRB 2003

5. Experimental results II
Kasumov et al., PRB 2003

6. Continuum elastic theory of a SWNT: Acoustic phonons
De Martino & Egger, PRB 2003
Displacement field:
Strain tensor:
Elastic energy density:
Suzuura & Ando, PRB 2002

7. Normal mode analysis
Breathing mode
Stretch mode
Twist mode

8. Electron-phonon coupling
Main contribution from deformation potential
couples to electron density
Other electron-phonon couplings small, but potentially responsible for Peierls distortion
Effective electron-electron interaction generated via phonon exchange (integrate out phonons)

9. SWNT as Luttinger liquid
Egger & Gogolin; Kane et al., PRL 1997
De Martino & Egger, PRB 2003
Low-energy theory of SWNT: Luttinger liquid
Coulomb interaction:
Breathing-mode phonon exchange causes attractive interaction:
Wentzel-Bardeen singularity: very thin SWNT
For (10,10) SWNT:

10. Superconductivity in ropes
De Martino & Egger, cond-mat/
Model:
Attractive electron-electron interaction within each of the N metallic SWNTs
Arbitrary Josephson coupling matrix, keep only singlet on-tube Cooper pair field
Single-particle hopping negligible Maarouf, Kane & Mele, PRB 2003

11. Order parameter for nanotube rope superconductivity
Hubbard Stratonovich transformation: complex order parameter field
to decouple Josephson terms
Integration over Luttinger liquid fields gives formally exact effective (Euclidean) action:

12. Quantum Ginzburg Landau (QGL) theory
1D fluctuations suppress superconductivity
Systematic cumulant & gradient expansion: Expansion parameter
QGL action, coefficients from full model

13. Amplitude of the order parameter
Mean-field transition at
For lower T, amplitudes are finite, with gapped fluctuations
Transverse fluctuations irrelevant for
QGL accurate down to very low T

14. Low-energy theory: Phase action
Fix amplitude at mean-field value: Low-energy physics related to phase fluctuations
Rigidity
from QGL, but also influenced by dissipation or disorder

15. Quantum phase slips: Kosterlitz-Thouless transition to normal state
Superconductivity can be destroyed by vortex excitations: Quantum phase slips (QPS)
Local destruction of superconducting order allows phase to slip by 2π
QPS proliferate for
True transition temperature

16. Resistance in superconducting state
QPS-induced resistance
Perturbative calculation, valid well below transition:

17. Comparison to experiment
Resistance below transition allows detailed comparison to Orsay experiments
Free parameters of the theory:
Interaction parameter, taken as
Number N of metallic SWNTs, known from residual resistance (contact resistance)
Josephson matrix (only largest eigenvalue needed), known from transition temperature
Only one fit parameter remains:

18. Comparison to experiment: Sample R2
Nice agreement
Fit parameter near 1
Rounding near transition is not described by theory
Quantum phase slips → low-temperature resistance
Thinnest known superconductors

19. Comparison to experiment: Sample R4
Again good agreement, but more noise in experimental data
Fit parameter now smaller than 1, dissipative effects
Ropes of carbon nanotubes thus allow to observe quantum phase slips

20. Josephson current through short tube
Buitelaar, Schönenberger
et al., PRL 2002, 2003
Short MWNT acts as (interacting) quantum dot
Superconducting reservoirs: Josephson current, Andreev conductance, proximity effect ?
Tunable properties (backgate), study interplay superconductivity ↔ dot correlations

21. Model
Short MWNT at low T: only a single spin-degenerate dot level is relevant
Anderson model
(symmetric)
Free parameters:
Superconducting gap ∆, phase difference across dot Φ
Charging energy U, with gate voltage tuned to single occupancy:
Hybridization Γ between dot and BCS leads

22. Supercurrent through nanoscale dot
How does correlated quantum dot affect the DC Josephson current?
Non-magnetic dot: Standard Josephson relation
Magnetic dot - Perturbation theory in Γ gives π-junction: Kulik, JETP 1965
Interplay Kondo effect – superconductivity?
Universality? Does only ratio matter?
Kondo temperature

23. Quantum Monte Carlo approach: Hirsch-Fye algorithm for BCS leads
Discretize imaginary time in stepsize
Discrete Hubbard-Stratonovich transformation → Ising field decouples Hubbard-U
Effective coupling strength:
Trace out lead & dot fermions → self-energies
Now stochastic sampling of Ising field

24. QMC approach Stochastic sampling of Ising paths
Siano & Egger
Stochastic sampling of Ising paths
Discretization error can be eliminated by extrapolation
Numerically exact results
Check: Perturbative results are reproduced
Low temperature, close to T=0 limit
Computationally intensive

25. Transition to π junction

26. Kondo regime to π junction crossover
Universality: Instead of Anderson parameters, everything controlled by ratio
Kondo regime has large Josephson current Glazman & Matveev, JETP 1989
Crossover to π junction at surprisingly large

27. Conclusions
Ropes of nanotubes exhibit intrinsic superconductivity, thinnest superconducting wires known
Low-temperature resistance allows to detect quantum phase slips in a clear way
Josephson current through short nanotube: Interplay between Kondo effect, superconductivity, and π junction